Ignition Delay Time Measurements Research Articles

Experimental and Kinetic Study on the Ignition Delay Times of 2,5-Dimethylfuran and the Comparison to 2-Methylfuran and Furan

The ignition phenomena of 2,5-dimethylfuran (DMF), 2-methylfuran (MF), and furan are systematically investigated. Ignition delay times are measured over the temperature range of 1150–2010 K and pressures of 1.2, 4, and 16 bar for lean, stoichiometric, and rich DMF/Ar/O2 mixtures. For comparison, similar measurements for stoichiometric MF/O2/Ar and furan/O2/Ar are also conducted. Through a multi-regression analysis, the measured ignition delay times of DMF are fitted empirically in an Arrhenius-like form as a function of experimental parameters. It is observed that, when the fuel concentration, pressure, temperature, and equivalence ratio are kept constant, furan has the longest ignition delay times, while the reactivity of DMF and MF strongly depends upon the temperature. The experimental ignition delay times are compared to model predictions of Somers et al. and Liu et al. Both models could give qualitative agreement with DMF and MF ignition data, while the model of Somers et al. provides better quantitative agreement. Modifications have been made to the model of Somers et al. to agree better with present ignition delay times and other sets of furan data. Further reaction pathway and sensitivity analysis are carried out to understand their combustion kinetics.

Experimental Study on Ethane Ignition Delay Times and Evaluation of Chemical Kinetic Models

The ignition delay times of ethane were measured using a high-pressure shock tube at different pressures (p = 1.2, 5.0, and 20.0 atm) and equivalence ratios (ϕ = 0.5, 1.0, and 2.0) with different argon diluent ratios. Correlations of the measured ignition delay times were provided. The measurements were compared to calculations from several representative chemical kinetic models to evaluate their performances. Results showed that Aramco Mech 1.3 could well reproduce the measured ignition delay times over a wide range, while GRI Mech 3.0 significantly overpredicted the measurements at stoichiometric and high equivalence ratio. To find out the reasons for the differences and similarities of the mechanisms on calculating the ignition delay time of ethane, sensitivity analysis and reaction pathway analysis were conducted. It is observed that the mechanisms have similar pathway for ethane consumption, while they have significant differences for ethyl radical decomposition reactions. Results also indicated that the incompleteness of the C2H5 + O2 reaction channels and the underestimation of the rate constant of reaction C2H4 + H (+M) = C2H5 (+M) might be responsible for the overestimation of GRI Mech 3.0 on ethane ignition. The mechanisms studied give similar prediction at a high equivalence ratio because reactions with a significant difference on rate constants do not show a high sensitivity coefficient at the condition.

Ignition delay times and chemical kinetics of diethoxymethane/O 2 /Ar mixtures

Abstract Autoignition properties of diethoxymethane/O2/Ar mixtures were systematically investigated using a shock-tube facility under various ignition conditions, including reflected shock pressures of 2−10 × 105 Pa, temperatures of 1065–1370 K, and equivalence ratios of 0.5, 1.0 and 2.0. Ignition delay times were determined by measuring the time interval between electronically excited CH∗ emission and reflected shock pressure signals monitored at shock-tube sidewall. An empirical correlation for ignition delay times as a function of temperature, pressure and equivalence ratio was deduced by linear regression analysis. A chemical kinetic mechanism describing the high-temperature oxidation of diethoxymethane was proposed. The theoretical predictions match experimental results quite well. Sensitivity and reaction pathway analyses were employed to identify dominant reactions and fuel consumption paths during the ignition process. To our knowledge, we are the first one to present the ignition delay time measurements of gas-phase diethoxymethane.

Effects of Bond Location on the Ignition and Reaction Pathways of trans-Hexene Isomers.

Chemical structure and bond location are well-known to impact combustion reactivity. The current work presents new experimental autoignition and speciation data on the three trans-hexene isomers (1-hexene, trans-2-hexene, and trans-3-hexene), which describe the effects of the location of the carbon-carbon double bond. Experiments were conducted with the University of Michigan rapid compression facility to determine ignition delay times from pressure time histories. Stoichiometric (ϕ = 1.0) mixtures at dilution levels of buffer gas:O2 = 7.5 (mole basis) were investigated at an average pressure of 11 atm and temperatures from 837 to 1086 K. Fast gas sampling and gas chromatography were also used to quantitatively measure 13 stable intermediate species formed during the ignition delay period of each isomer at a temperature of ∼900 K. The measured ignition delay times and species measurements were in good agreement with previous experimental studies at overlapping conditions. The results were modeled using a gasoline surrogate reaction mechanism from Lawrence Livermore National Laboratory, which contains a submechanism for the trans-hexene isomers. The model predictions captured the overall autoignition characteristics of the hexene isomers well (within a factor of 2), as well as the time histories of several of the intermediate species (e.g., propene). However, there were discrepancies between the model predictions and the experimental data for some species, particularly for the 3-hexene isomer.

Comparative Study on Autoignition Characteristics of Methylcyclohexane and Cyclohexane

Ignition delay times were measured behind reflected shock waves for cyclohexane and methylcyclohexane at pressures of 1.1, 5.0, and 16.0 atm, temperatures from 1075 to 1750 K, and equivalence ratios of 0.5, 1.0, and 2.0. Correlations of the ignition delay times were performed at the three equivalence ratios in Arrhenius form. Measured ignition delay times showed fairly good agreements with previous data. Several accepted mechanisms (JetSurF 2.0, Wang et al., Sirjean et al., Orme et al., and Silke et al.) were used to simulate the experimental measurements and conduct flux analyses and sensitivity analyses. Comparisons of the simulations and analyses between the mechanisms give insights into the oxidation of methylcyclohexane and cyclohexane. Methylcyclohexane has an evidently longer ignition delay time than cyclohexane at ϕ = 0.5, while its ignition delay time becomes comparative to those of cyclohexane at ϕ = 1.0. Chemical kinetic interpretation is given for this observation.

Kinetic modeling study of hydrogen addition effects on ignition characteristics of dimethyl ether at engine-relevant conditions

The effects of hydrogen on ignition characteristics of dimethyl ether (DME) with air under engine-relevant conditions were studied by CHEMKIN package coupled with chemical kinetic models. Firstly, four recently developed DME models were used to validate the measured ignition delay times from previous literature. The validation results showed that NUIG Mech_56.54 agrees the literature data best and thus was chosen to conduct the following simulations and chemical kinetic analysis of ignition behaviors of DME/H2 mixture at low temperature. Subsequently, simulations of auto-ignition of DME/H2/air using NUIG Mech_56.54 were made under various hydrogen blending ratios (0, 50%, 80%, 90%, 95%, 98% and 100%). It was found that hydrogen retarded the ignition timing of DME at engine-relevant conditions and this effect was non-linear with the increase of hydrogen blending ratio. The inhibition and non-linear effect was chemically interpreted by radical mole fraction analysis. Finally, reaction pathway and sensitivity analysis were made to gain further insight on ignition chemistries of DME/H2 blends under engine-relevant conditions.

Shock tube and modeling study of 2,7-dimethyloctane pyrolysis and oxidation

High molecular weight iso-paraffinic molecules are found in conventional petroleum, Fischer–Tropsch (FT), and other alternative hydrocarbon fuels, yet fundamental combustion studies on this class of compounds are lacking. In the present work, ignition delay time measurements in 2,7-dimethyloctane/air were carried out behind reflected shock waves using conventional and constrained reaction volume (CRV) methods. The ignition delay time measurements covered the temperature range 666–1216K, pressure range 12–27atm, and equivalence ratio of 0.5 and 1. The ignition delay time temperatures span the low-, intermediate- and high-temperature regimes for 2,7-dimethyloctane (2,7-DMO) oxidation. Clear evidence of negative temperature coefficient behavior was observed near 800K. Fuel time-history measurements were also carried out in pyrolysis experiments in mixtures of 2000ppm 2,7-DMO/argon at pressures near 16 and 35atm, and in the temperature range of 1126–1455K. Based on the fuel removal rates, the overall 2,7-DMO decomposition rate constant can be represented with k=4.47×105 exp(−23.4[kcal/mol]/RT) [1/s]. Ethylene time-history measurements in pyrolysis experiments at 16atm are also provided. The current shock tube dataset was simulated using a novel chemical kinetic model for 2,7-DMO. The reaction mechanism includes comprehensive low- and high-temperature reaction classes with rate constants assigned using established rules. Comparisons between the simulated and experimental data show simulations reproduce the qualitative trends across the entire range of conditions tested. However, the present kinetic modeling simulations cannot quantitatively reproduce a number of experimental data points, and these are analyzed herein.

317 急速圧縮装置による実用ガソリンの着火遅れ時間計測(OS3-(4),OS3 オーガナイズドセッション≪熱・流体・エネルギー変換工学の進展≫)

317 急速圧縮装置による実用ガソリンの着火遅れ時間計測(OS3-(4),OS3 オーガナイズドセッション≪熱・流体・エネルギー変換工学の進展≫)

Autoignition of propane behind shock waves

The ignition delay times of C3H8-O2-Ar mixtures of various compositions are measured behind reflected shock waves at temperatures of 1220–1750 K and a pressure of (0.65 ± 0.05) MPa. A kinetic mechanism of propane ignition based on the known rate constants of the key elementary reactions is developed. The scheme closely describes our own and published experimental data on the ignition of propane-oxygen mixtures behind shock waves over a wide ranges of temperatures (1100–1750 K), pressures (0.12–0.65 MPa), and propane (0.05–11%) and oxygen (0.25–21%) concentrations. At T < 1100 K, neither the proposed kinetic scheme nor the different kinetic mechanisms reported in the literature can satisfactorily describe the experimentally measured ignition delay times for propane behind shock waves. We suggested that, at T < 1100 K, the limiting process is the propagation of deflagration flames from individual C3H8 ignition hotspots. Numerical calculations of the propagation of a one-dimensional flame performed using the proposed kinetic scheme qualitatively confirm this assumption.

Shock Tube and Kinetic Modeling Study of Cyclopentane and Methylcyclopentane

Ignition delay times for 1% cyclopentane/O2 and 0.833% methylcyclopentane/O2 mixtures diluted by argon were measured behind reflected shock waves at pressures of 1.1 and 10 atm, with equivalence ratios of 0.577, 1.0, and 2.0, and in the temperature range from 1150 to 1850 K. Submechanisms for cyclopentane and methylcyclopentane were developed and added to the JetSurF2.0 mechanism for the kinetic interpretation of cyclopentane and methylcyclopentane oxidation chemistry at the high temperature region. Simulations with the model exhibit fairly good agreements with the measured ignition delay times of both cyclopentane and methylcyclopentane under all tested conditions. Cyclopentane shows longer ignition delay time than methylcyclopentane, especially for the fuel-lean mixture. Reaction pathways and sensitivity analyses were conducted to get insights into the oxidation process of cyclopentane and methylcyclopentane. Then, three factors are given for the effect of a cyclic ring and substitution of a methyl group. Substitution of a methyl group weakens the C–C bond to motivate fuel unimolecular decomposition. The shape of the cyclic ring determines the chain alkyl radicals, affecting regeneration and accumulation of H radical. The presence of a methyl group also leads to different alkyl radicals.

Experimental and Kinetic Modeling Study of Ignition Characteristics of Chinese RP-3 Kerosene

The ignition delay times of gas-phase Chinese RP-3 kerosene/O2/Ar mixtures were measured by monitoring pressure and electronically excited OH chemiluminescence signals behind reflected shock waves in a chemical shock tube. The experiments were performed over the temperature range of 1100–1600 K, at pressures of 0.1 MPa, 0.2 MPa, and 0.3 MPa, and for equivalence ratios of 0.5, 1.0, and 1.5. The effects of temperature, pressure, and equivalence ratio on the ignition delay time were also investigated. The results show that a linear relationship exists between the reciprocal of temperature and the logarithm of the ignition delay times, and an increase in temperature results in a decrease in the measured ignition delay times at all conditions. Furthermore, with equivalence ratio decreasing or pressure increasing, the measured ignition delay times are decreasing. A new surrogate fuel consists of n-decane (65 vol%), toluene (10 vol%), and propylcyclohexane (25 vol%) for Chinese RP-3 kerosene was presented and a reaction mechanism consists of 150 chemical species and 591 elemental reactions of this surrogate fuel was developed. Comparisons of the calculated ignition delay times of this surrogate fuel using this reaction mechanism with the experimental ignition delay times of Chinese RP-3 kerosene show excellent agreement.

Shock Tube Study of Dimethylamine Oxidation

ABSTRACTDimethylamine (DMA) ignition delay times and OH time histories during the oxidation process were investigated behind reflected shock waves. The ignition delay time measurements cover the temperature range of 1181–1498 K, with pressures near 0.9, 1.5, and 2.8 atm, and equivalence ratios of 0.5, 1, and 2, in 4% oxygen/argon. The ignition delay time data feature low scatter and can be correlated to a single expression with 2 ∼ 0.99: τign = 7.30 × 10−4 −0.68 Φ0.45 exp(18,265/), where τign is in μs, in atm, and in K. OH time histories were measured using laser absorption of the R1(5) line of the A‐X(0,0) transition near 306.7 nm, in stoichiometric mixtures of 500 ppm DMA/O2/argon. The mechanism developed by Li et al. was used initially to simulate the measured DMA ignition delay times and the OH time histories. The Li et al. mechanism was then updated by adding the DMA unimolecular decomposition channel: DMA = CH3NH + CH3, with the reaction rate constant estimated by analogy to dimethyl ether decomposition, previously investigated by Cook et al. The reactions of DMA + OH were also updated based on recent work in our laboratory. The simulation results using the modified Li et al. mechanism are in good agreement with both the ignition delay times and OH time‐history data.

Effect of pressure and equivalence ratio on the ignition characteristics of dimethyl ether-hydrogen mixtures

Experimental and numerical study on the effect of pressure and equivalence ratio on the ignition delay times of the DME/H2/O2 mixtures diluted in argon were conducted using a shock tube and CHEMKIN II package at equivalence ratios of 0.5–2.0, pressures of 1.2–10 atm and hydrogen fractions of 0–100%. It was found that the measured ignition delay times of the DME/H2 mixtures demonstrate three ignition regimes. For the DME/H2 mixture at XH2 ≤80%, the ignition is controlled by the DME chemistry and ignition delay times present a typical Arrhenius pressure dependence and weak equivalence ratio dependence. For the DME/H2 mixture at 80% < XH2 < 98%, the ignition is controlled by the combined chemistries of DME and hydrogen, and the ignition delay times give higher ignition activation energy at higher pressures and a typical Arrhenius equivalence ratio dependence. However, for the DME/H2 mixture at XH2≥98%, the ignition is controlled by the hydrogen chemistry and ignition delay time shows complex pressure dependence and weak equivalence ratio dependence. Comparison of the measurements of neat DME and neat hydrogen with the calculations using three generally accepted mechanisms, NUIG Aramco Mech 1.3 [1], LLNL DME Mech [2–4] and Princeton-Zhao Mech [5], shows that NUIG Aramco Mech 1.3 gives the best predictions and can well capture the pressure and equivalence ratio dependence at various hydrogen fractions. The sensitivity and normalized H-radicals consumption analysis were performed using NUIG Aramco Mech 1.3 and the key reactions that control the ignition characteristics of DME/H2 mixtures were revealed. Further chemical kinetic analysis was made to interpret the ignition delay time dependence on pressure and equivalence ratio at varied hydrogen fractions.

Experimental and modeling study on the high-temperature oxidation of Ammonia and related NOx chemistry

Ammonia oxidation and ignition delay time measurements for pressures above 10atm are scarce. In addition, NH3 is known to adsorb on stainless steel, so measurement results could be in question if wall passivation is not employed for apparatuses utilizing steel. To overcome these measurement difficulties and overall lack of high-pressure data for ammonia, new and methodical ignition delay time measurements have been performed behind reflected shock waves over a wide range of temperatures (1560–2455K), pressures (around 1.4, 11, and 30atm) and equivalence ratios (0.5, 1.0, and 2.0) for mixtures of ammonia highly diluted in Ar (98–99%). The new set of data from the present study was compared to several models from the literature. It was found that a large majority of the models do not predict the ignition delay times with accuracy, and there is a surprisingly wide variation amongst the predictions. One satisfactory model, from Dagaut et al. (2008), was selected and extended to compounds other than NH3 using H2/O2/CO, N2O, NO2, and NNH sub-mechanisms from the literature. The resulting comprehensive mechanism predicts well the ammonia ignition delay time data from the present study along with other NH3, NO2, and N2O data from the authors as well as from the literature with high accuracy. In addition to the new ammonia oxidation data and related model comparisons, the present paper documents a state-of-the-art NOx sub-mechanism that can be used for a wide range of combustion calculations when added to, for example, baseline mechanisms involving hydrogen and hydrocarbon kinetics.

An ignition delay and kinetic modeling study of methane, dimethyl ether, and their mixtures at high pressures

The development of accurate chemical kinetic models capable of predicting the combustion of methane and dimethyl ether in common combustion environments such as compression ignition engines and gas turbines is important as it provides valuable data and understanding of these fuels under conditions that are difficult and expensive to study in the real combustors. In this work, both experimental and chemical kinetic model-predicted ignition delay time data are provided covering a range of conditions relevant to gas turbine environments (T=600–1600K, p=7–41atm, ϕ=0.3, 0.5, 1.0, and 2.0 in ‘air’ mixtures). The detailed chemical kinetic model (Mech_56.54) is capable of accurately predicting this wide range of data, and it is the first mechanism to incorporate high-level rate constant measurements and calculations where available for the reactions of DME. This mechanism is also the first to apply a pressure-dependent treatment to the low-temperature reactions of DME. It has been validated using available literature data including flow reactor, jet-stirred reactor, shock-tube ignition delay times, shock-tube speciation, flame speed, and flame speciation data. New ignition delay time measurements are presented for methane, dimethyl ether, and their mixtures; these data were obtained using three different shock tubes and a rapid compression machine. In addition to the DME/CH4 blends, high-pressure data for pure DME and pure methane were also obtained. Where possible, the new data were compared with existing data from the literature, with good agreement.

An experimental and modeling study of propene oxidation. Part 2: Ignition delay time and flame speed measurements

Experimental data obtained in this study (Part II) complement the speciation data presented in Part I, but also offer a basis for extensive facility cross-comparisons for both experimental ignition delay time (IDT) and laminar flame speed (LFS) observables.To improve our understanding of the ignition characteristics of propene, a series of IDT experiments were performed in six different shock tubes and two rapid compression machines (RCMs) under conditions not previously studied. This work is the first of its kind to directly compare ignition in several different shock tubes over a wide range of conditions. For common nominal reaction conditions among these facilities, cross-comparison of shock tube IDTs suggests 20–30% reproducibility (2σ) for the IDT observable. The combination of shock tube and RCM data greatly expands the data available for validation of propene oxidation models to higher pressures (2–40atm) and lower temperatures (750–1750K).Propene flames were studied at pressures from 1 to 20atm and unburned gas temperatures of 295–398K for a range of equivalence ratios and dilutions in different facilities. The present propene–air LFS results at 1atm were also compared to LFS measurements from the literature. With respect to initial reaction conditions, the present experimental LFS cross-comparison is not as comprehensive as the IDT comparison; however, it still suggests reproducibility limits for the LFS observable. For the LFS results, there was agreement between certain data sets and for certain equivalence ratios (mostly in the lean region), but the remaining discrepancies highlight the need to reduce uncertainties in laminar flame speed experiments amongst different groups and different methods. Moreover, this is the first study to investigate the burning rate characteristics of propene at elevated pressures (>5atm).IDT and LFS measurements are compared to predictions of the chemical kinetic mechanism presented in Part I and good agreement is observed.

A ‘frozen electric-field’ approach to simulate repetitively pulsed nanosecond plasma discharges and ignition of hydrogen–air mixtures

High-fidelity modelling of nanosecond repetitively pulsed discharges (NRPDs) is burdened by the multiple time and length scales and large chemistry mechanisms involved, which prohibit detailed analyses and parametric studies. In the present work, we propose a ‘frozen electric-field’ modelling approach to expedite the NRPD simulations without adverse effects on the solution accuracy. First, a burst of nanosecond voltage pulses is simulated self-consistently until the discharge reaches a stationary state. The calculated spatial distributions and temporal evolution of the electric field, electron density and electron energy during the last pulse are then stored in a library and the electrical characteristics of subsequent pulses are frozen at these values. This strategy allows the timestep for numerical integration to be increased by four orders of magnitude (from 10−13 to 10−9 s), thereby significantly improving the computational efficiency of the process. Reduced calculations of a burst of 50 discharge pulses show good agreement with the predictions from a complete plasma model (electrical characteristics calculated during each pulse). The error in species densities is less than 20% at the centre of the discharge volume and about 30% near the boundaries. The deviations in temperature, however, are much lower, at 5% in the entire domain. The model predictions are in excellent agreement with measured ignition delay times and temperatures in H2–air mixtures subject to dielectric barrier NRPD over a pressure range of 54–144 Torr with equivalence ratios of 0.7–1.2. The OH density increases with pressure and triggers low-temperature fuel oxidation, which leads to rapid temperature rise and ignition. The ignition delay decreases by a factor of 2, with an increase in pressure from 54 to 144 Torr. In contrast, an increase in the H2–air equivalence ratio from 0.7 to 1.2 marginally decreases the ignition delay by about 20%. This behaviour is attributed to the insensitivity of OH production rates to the variation in the equivalence ratio.

Shock-Tube Study on Ethylcyclohexane Ignition

Ignition delay times of ethylcyclohexane with a fixed fuel concentration of 0.5% were measured behind reflected shock wave at pressures of 1.1–10.0 atm, equivalence ratios of 0.5–2.0, and temperatures between 1000 and 1700 K. The measured ignition delay times were compared to predictions by the mechanism JetSurF2.0, and fairly good agreements were presented under the test conditions. The ignition delay time of the fuel-rich mixture was suggested to overwhelm that of the fuel-lean mixture when tested at a low temperature and an elevated pressure. Analysis was conducted to kinetically interpret the effects of the pressure, temperature, equivalence ratio, and fuel concentration on the time histories of the OH radical and intermediate species of hydrocarbons. Results showed that the suppression of reaction H + O2 ⇄ OH + O by fuel-concerned radicals played an important role. Sensitivity analyses were carried out to provide insight into the controlling reactions on the ignition of ethylcyclohexane.

Ignition delay time measurements behind reflected shock-waves for a representative coal-derived syngas with and without NH3 and H2S impurities

The composition of a representative coal-derived syngas was determined by averaging 40 practical coal syngas compositions from the literature and corresponds to a departure from many recent studies which only focus on syngas blends containing just CO and H2. Ignition delay times have been measured behind reflected shock waves for this averaged mixture with an equivalence ratio of 0.5 (0.4554% CO/0.3297% H2/0.1032% CO2/0.0172% CH4/0.2407% H2O/0.8538% O2 in 98% Ar (mol.%)) at around 1.7, 13, and 32atm. The same mixture was also investigated with impurities (200ppm of NH3 and 50ppm of H2S). Care was taken when working with the blends containing H2O and NH3 to avoid errors in the shock-tube composition; direct measurement of the water vapor mole fractions were performed using a tunable diode laser absorption diagnostic near 1.38μm. The effect of the various constituents on the ignition delay time was also investigated by comparing to results from a baseline mixture (H2/CO/O2/Ar) and results with this baseline mixture with only one of the other constituents of the syngas (i.e., CO2, CH4, H2S). Experimental data were compared with recent detailed kinetics mechanisms from the literature. Results showed that, under the conditions of this study, extending the mixture composition to include realistic concentrations of species beyond just the CO and H2 does not have a very large effect on the ignition delay time for a coal-derived syngas. However, a comparison of this coal-derived syngas with a syngas derived from biomass, tested in an earlier study by the authors, exhibited large differences due to the larger CH4 concentration in the bio-derived syngas. Two chemical kinetic models from the literature were found suitable to reproduce these data over most of the range of mixtures, temperatures, and pressures investigated, namely the mechanisms associated with Galway and with Princeton.

The high-temperature autoignition of biodiesels and biodiesel components

Ignition delay time measurements are reported for two reference fatty-acid methyl ester biodiesel fuels, derived from methanol-based transesterification of soybean oil and animal fats, and four primary constituents of all methyl ester biodiesels: methyl palmitate, methyl stearate, methyl oleate, and methyl linoleate. Experiments were carried out behind reflected shock waves for gaseous fuel/air mixtures at temperatures ranging from 900 to 1350K and at pressures around 10 and 20atm. Ignition delay times were determined by monitoring pressure and ultraviolet chemiluminescence from electronically-excited OH radicals. The results show similarity in ignition delay times for all methyl ester fuels considered, irrespective of the variations in organic structure, at the high-temperature conditions studied and also similarity in high-temperature ignition delay times for methyl esters and n-alkanes. Comparisons with recent kinetic model efforts are encouraging, showing deviations of at most a factor of two and in many cases significantly less.